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Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets

Authors Ángel Javier Alonso , Michael Kerber



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Author Details

Ángel Javier Alonso
  • Technische Universität Graz, Austria
Michael Kerber
  • Technische Universität Graz, Austria

Acknowledgements

The authors thank Jan Jendrysiak for helpful discussions. We are also grateful to the anonymous reviewers for their careful reading of our manuscript and their detailed comments and suggestions.

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Ángel Javier Alonso and Michael Kerber. Decomposition of Zero-Dimensional Persistence Modules via Rooted Subsets. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 7:1-7:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SoCG.2023.7

Abstract

We study the decomposition of zero-dimensional persistence modules, viewed as functors valued in the category of vector spaces factorizing through sets. Instead of working directly at the level of vector spaces, we take a step back and first study the decomposition problem at the level of sets. This approach allows us to define the combinatorial notion of rooted subsets. In the case of a filtered metric space M, rooted subsets relate the clustering behavior of the points of M with the decomposition of the associated persistence module. In particular, we can identify intervals in such a decomposition quickly. In addition, rooted subsets can be understood as a generalization of the elder rule, and are also related to the notion of constant conqueror of Cai, Kim, Mémoli and Wang. As an application, we give a lower bound on the number of intervals that we can expect in the decomposition of zero-dimensional persistence modules of a density-Rips filtration in Euclidean space: in the limit, and under very general circumstances, we can expect that at least 25% of the indecomposable summands are interval modules.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Topology
  • Theory of computation → Computational geometry
Keywords
  • Multiparameter persistent homology
  • Clustering
  • Decomposition of persistence modules
  • Elder Rule

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